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Common Unit Circle Values

A free Trigonometry lesson from the “The Unit Circle” unit, with a worked example and practice problems including step-by-step solutions.

Common values become predictable when students combine special angles, coordinates, and quadrant signs instead of memorizing a chart blindly. In this lesson, the goal is to recognize the most common exact sine, cosine, and tangent values. Prerequisite check: Algebra II or College Algebra foundations. Example 1: at pi/3, the unit-circle point is (1/2, sqrt(3)/2), so cos(pi/3) = 1/2 and sin(pi/3) = sqrt(3)/2. Example 2: at 7pi/6, the reference angle is pi/6 and both sine and cosine are negative. A common mistake is memorizing a value without checking its sign; the safer habit is to find the reference angle, choose the special value, then apply the quadrant sign.

What you'll learn

Why it matters: The unit circle explains signals, waves, rotations, and why trig values repeat instead of being isolated facts.

Worked example

Problem. Example 1 Foundation: What does the unit-circle coordinate (cos(theta), sin(theta)) help you find?

  1. Cosine is x and sine is y.
  2. The unit circle turns trig values into coordinates.
  3. That connection powers graphing and equations.

Answer: sine and cosine values for the angle

Practice problems

1. Practice 1 Foundation: What unit-circle point matches pi/4?

Choices: (sqrt(2)/2, sqrt(2)/2) · (1, 0) · (0, 1) · (-1, 0)

Show solution
  1. Use the reference angle and quadrant.
  2. The point for pi/4 is (sqrt(2)/2, sqrt(2)/2).
  3. The x-coordinate is cosine and the y-coordinate is sine.

Answer: (sqrt(2)/2, sqrt(2)/2)

2. Practice 2 Setup: Find sin(2pi/3).

Choices: sqrt(3)/2 · -1/2 · -sqrt(3) · 1

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  1. Warm-up: First identify exactly what the question is asking: Practice 2 Setup: Find sin(2pi/3).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Sine is the y-coordinate.
  4. The point is (-1/2, sqrt(3)/2).
  5. So sin(2pi/3) = sqrt(3)/2.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: sqrt(3)/2

3. Practice 3 Meaning: Find cos(7pi/6).

Choices: -sqrt(3)/2 · -1/2 · sqrt(3)/3 · -1

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice 3 Meaning: Find cos(7pi/6).
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  3. Cosine is the x-coordinate.
  4. The point is (-sqrt(3)/2, -1/2).
  5. So cos(7pi/6) = -sqrt(3)/2.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: -sqrt(3)/2

4. Practice 4 Method: Where does 3pi/2 land?

Choices: negative y-axis · Quadrant I · Quadrant IV · positive x-axis

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice 4 Method: Where does 3pi/2 land?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Locate the terminal side.
  4. 3pi/2 lands in negative y-axis.
  5. The location controls the signs.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: negative y-axis

5. Practice 5 Reasoning: Find tan(pi/6).

Choices: sqrt(3)/3 · 1/2 · sqrt(3)/2 · 0

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice 5 Reasoning: Find tan(pi/6).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Tangent is sine divided by cosine.
  4. Using (sqrt(3)/2, 1/2), tan(pi/6) = sqrt(3)/3.
  5. If cosine is 0, tangent is undefined.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: sqrt(3)/3

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